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Register a new userProbabilistic Graphical Models and Tensor Networks: A Hybrid Framework
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We investigate a correspondence between two formalisms for discrete probabilistic modeling: probabilistic graphical models (PGMs) and tensor networks (TNs), a powerful modeling framework for simulating complex quantum systems. The graphical calculus of PGMs and TNs exhibits many similarities, with discrete undirected graphical models (UGMs) being a special case of TNs. However, more general probabilistic TN models such as Born machines (BMs) employ complex-valued hidden states to produce novel forms of correlation among the probabilities. While representing a new modeling resource for capturing structure in discrete probability distributions, this behavior also renders the direct application of standard PGM tools impossible. We aim to bridge this gap by introducing a hybrid PGM-TN formalism that integrates quantum-like correlations into PGM models in a principled manner, using the physically-motivated concept of decoherence. We first prove that applying decoherence to the entirety of a BM model converts it into a discrete UGM, and conversely, that any subgraph of a discrete UGM can be represented as a decohered BM. This method allows a broad family of probabilistic TN models to be encoded as partially decohered BMs, a fact we leverage to combine the representational strengths of both model families. We experimentally verify the performance of such hybrid models in a sequential modeling task, and identify promising uses of our method within the context of existing applications of graphical models.
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Optimal Transport (OT) is being widely used in various fields such as machine learning and computer vision, as it is a powerful tool for measuring the similarity between probability distributions and histograms. In previous studies, OT has been defined as the minimum cost to transport probability mass from one probability distribution to another. In this study, we propose a new framework in which OT is considered as a maximum a posteriori (MAP) solution of a probabilistic generative model. With the proposed framework, we show that OT with entropic regularization is equivalent to maximizing a posterior probability of a probabilistic model called Collective Graphical Model (CGM), which describes aggregated statistics of multiple samples generated from a graphical model. Interpreting OT as a MAP solution of a CGM has the following two advantages: (i) We can calculate the discrepancy between noisy histograms by modeling noise distributions. Since various distributions can be used for noise modeling, it is possible to select the noise distribution flexibly to suit the situation. (ii) We can construct a new method for interpolation between histograms, which is an important application of OT. The proposed method allows for intuitive modeling based on the probabilistic interpretations, and a simple and efficient estimation algorithm is available. Experiments using synthetic and real-world spatio-temporal population datasets show the effectiveness of the proposed interpolation method.
We develop a probabilistic framework for deep learning based on the Deep Rendering Mixture Model (DRMM), a new generative probabilistic model that explicitly capture variations in data due to latent task nuisance variables. We demonstrate that max-sum inference in the DRMM yields an algorithm that exactly reproduces the operations in deep convolutional neural networks (DCNs), providing a first principles derivation. Our framework provides new insights into the successes and shortcomings of DCNs as well as a principled route to their improvement. DRMM training via the Expectation-Maximization (EM) algorithm is a powerful alternative to DCN back-propagation, and initial training results are promising. Classification based on the DRMM and other variants outperforms DCNs in supervised digit classification, training 2-3x faster while achieving similar accuracy. Moreover, the DRMM is applicable to semi-supervised and unsupervised learning tasks, achieving results that are state-of-the-art in several categories on the MNIST benchmark and comparable to state of the art on the CIFAR10 benchmark.
Probabilistic graphical models, such as Markov random fields (MRF), exploit dependencies among random variables to model a rich family of joint probability distributions. Sophisticated inference algorithms, such as belief propagation (BP), can effectively compute the marginal posteriors. Nonetheless, it is still difficult to interpret the inference outcomes for important human decision making. There is no existing method to rigorously attribute the inference outcomes to the contributing factors of the graphical models. Shapley values provide an axiomatic framework, but naively computing or even approximating the values on general graphical models is challenging and less studied. We propose GraphShapley to integrate the decomposability of Shapley values, the structure of MRFs, and the iterative nature of BP inference in a principled way for fast Shapley value computation, that 1) systematically enumerates the important contributions to the Shapley values of the explaining variables without duplicate; 2) incrementally compute the contributions without starting from scratches. We theoretically characterize GraphShapley regarding independence, equal contribution, and additivity. On nine graphs, we demonstrate that GraphShapley provides sensible and practical explanations.
We introduce a graphical user interface for constructing arbitrary tensor networks and specifying common operations like contractions or splitting, denoted GuiTeNet. Tensors are represented as nodes with attached legs, corresponding to the ordered dimensions of the tensor. GuiTeNet visualizes the current network, and instantly generates Python/NumPy source code for the hitherto sequence of user actions. Support for additional programming languages is planned for the future. We discuss the elementary operations on tensor networks used by GuiTeNet, together with high-level optimization strategies. The software runs directly in web browsers and is available online at http://guitenet.org.
Non-negative matrix and tensor factorisations are a classical tool for finding low-dimensional representations of high-dimensional datasets. In applications such as imaging, datasets can be regarded as distributions supported on a space with metric structure. In such a setting, a loss function based on the Wasserstein distance of optimal transportation theory is a natural choice since it incorporates the underlying geometry of the data. We introduce a general mathematical framework for computing non-negative factorisations of both matrices and tensors with respect to an optimal transport loss. We derive an efficient computational method for its solution using a convex dual formulation, and demonstrate the applicability of this approach with several numerical illustrations with both matrix and tensor-valued data.
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